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In representation theory, the notion of twisted intertwiners is a generalization of that of intertwiners.

References using the exact terminology “twisted intertwiner” include Fuchs & Schweigert 2000a (7.2), 2000b (2.2), Felder, Fröhlich, Fuchs & Schweigert 2000 (4.7), but the notion itself is older and may be known under other names.

Definition

Definition

For GG a group and ρ 1,ρ 2Rep(G)\rho_1, \rho_2 \;\in\; Rep(G) a pair of linear representation on vector spaces V 1V_1, V 2V_2, respectively (all this generalizes straightforwardly to algebras and their modules), a twisted intertwiner

ρ 1ρ 2 \rho_1 \Rightarrow \rho_2

is

  1. an automorphism αAut(G)\alpha \in Aut(G) of the group

  2. a linear map η:V 1V 2\eta \colon V_1 \longrightarrow V_2

such that for all gGg \in G we have

ηρ 1(g)=ρ 2(α(g))η. \eta \circ \rho_1(g) \;=\; \rho_2\big(\alpha(g)\big) \circ \eta \,.

Given a pair of twisted intertwiners ρ 1(α,η)ρ 2overst(α,η)ρ 3\rho_1 \overset{(\alpha,\eta)}{\Rightarrow} \rho_2 \overst{(\alpha',\eta')}{\Rightarrow} \rho_3 their composition is given by composing their components separately.

Properties

Category-theoretically

Writing BG\mathbf{B}G for the delooping groupoid of GG and VecVec for the category of vector spaces (over a given ground field), the ordinary category Rep of GG-representations and ordinary intertwiners η:ρ 1ρ 2\eta \colon \rho_1 \Rightarrow \rho_2 between these is equivalently the functor category

Rep(G)Func(BG,Vec). Rep(G) \;\;\simeq\;\; Func\big( \mathbf{B}G ,\, Vec \big) \,.

In contrast, the category of GG-representations with twisted intertwiners between them is the iso-comma (2,1)-category between BG:BAut(G)Grpd\mathbf{B}G \colon \mathbf{B}Aut(G) \longrightarrow Grpd and Vec:*GrpdVec \colon \ast \to Grpd, whose 1-morphisms are diagrams in Grpd of this form:

and whose 2-morphisms are the evident paper-cup pasting diagrams

References

Particle-hole symmetry.

Original

>1\neq \gt 1 may be understood as ν=1\nu = 1 for majority spin polarization with ν1\nu - 1 for the minority polarization (Haldane 1983 doi:10.1103/PhysRevLett.51.605)

Here

  • \bullet” means that the corresponding combination of (k,q)(k,q) is not admissible, in that kqk q is odd or the Gauss sum n=0 k1e πikqn 2=0\sum_{n=0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} q n^2 } = 0.

  • \circ” means that qk>2\tfrac{q}{k} \gt 2

So:

  1. in the column of q=1q = 1 all odd kk are excluded and all even kk are admissible,

  2. in the column of q=2q = 2 all odd kk are excluded and exactly every second even kk is admissible,

  3. in rows of odd kk the odd qq are excluded, and for even q=2rq = 2 r the result is admissible iff the Jacobi symbol (r|k)0(r \vert k) \neq 0.

    Since the Jacobi symbol (r|k)(r \vert k) vanishes iff gcd(r,k)1gcd(r,k) \neq 1, this means that, at least for odd kk, exactly only the reduced fractions appear.

Last revised on April 2, 2025 at 17:14:41. See the history of this page for a list of all contributions to it.